2010/06/30 by Richard Hain
Mathematics · #math.AG #msc:14D05 #msc:14H15 #msc:14F35 #msc:14F45
paper · pdf · doi:10.1215/00127094-1593299
published as Duke Math. J. 161, no. 7 (2012), 1351-1378 · Updated to the final version. This is equivalent to the published version with the published correction. Also added journal reference
arxiv created 2013/10/19 · arxiv updated 2019/12/19
Suppose that g > 2, that n > 0 and that m > 0. In this paper we show that if E is an irreducible smooth variety which dominates a divisor D in Mg,n[m], the moduli space of n-pointed, smooth curves of genus g with a level m structure, then the closure of the image of the monodromy representation pi1(E,e)--> Spg(Zhat) has finite index in Spg(Zhat). A similar result is proved for codimension 1 families of the universal principally polarized abelian variety of dimension g > 2. Both results are deduced from a general "non-abelian strictness theorem". The first result is used in arXiv:1001.5008 to control the Galois cohomology of the function field of Mg,n[m] in degrees 1 and 2.